SPECIAL TYPES OF INTERVAL VALUED NEUTROSOPHIC GRAPHS

SPECIAL TYPES OF INTERVAL VALUED NEUTROSOPHIC GRAPHS M. A. MALIK1 , A. HASSAN2 Abstract. Neutrosophic theory has many applications in graph theory, interval valued neutrosophic graph (IVNG) is the generalization of fuzzy graph, intuitionistic fuzzy graph and single valued neutrosophic graph. In this paper, we introduced some types of IVNGs, which are subdivision IVNGs, middle IVNGs, total IVNGs and interval valued neutrosophic line graphs (IVNLGs), also discussed the isomorphism, co weak isomorphism and weak isomorphism properties of subdivision IVNGs, middle IVNGs, total IVNGs and IVNLGs. Keywords: Interval valued neutrosophic line graph, Subdivision IVNG, middle IVNG, total IVNG.

1. Introduction Neutrosopic sets were introduced by Smarandache in [1], which are the generalization of fuzzy sets and intuitionistic fuzzy sets. The single valued neutrosophic graphs were introduced by Broumi, Talea, Bakali and Smarandache in [3] and recently in [8, 9, 10] proposed some algorithms. A graph is a way to represent information between objects.

The objects are represented by vertices and the relations by edges. When there is vagueness in the description of the objects or in its relationships or in both, it is natural that we need to design a fuzzy graph Model. The perception of fuzzy graph was introduced by Rosenfeld in [6] and the some remarks on fuzzy graphs were explained by Bhattacharya in [5]. The special types and its truncations of fuzzy graphs were paid the way by Gani in [7]. The IVNGs have many applications in path problems, networks and computer science.

The strong IVNG and complete IVNG are the special types of IVNG. In this paper, we introduce the another types of IVNGs, which are subdivision IVNGs, middle IVNGs, total IVNGs and IVNLGs. These are all the strong IVNGs, also we discuss their relations based on isomorphism, co weak isomorphism and weak isomorphism.

2. Preliminaries Let G denotes IVNG and G∗ = (V, E) denotes its underlying crisp graph.

1

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. e-mail: malikpu@yahoo.com ;

2

alihassan.iiui.math@gmail.com; .

1

2

Definition 2.1. [1, 2] Let X be a crisp set, the single (interval) valued neutrosophic set (SVNS) A is characterized by three membership functions T A (x), IA (x) and FA (x), such that for every x ∈ X, the membership values T A (x), IA (x), FA (x) ∈ [0, 1] (TA (x), IA (x), FA (x) ⊆ [0, 1].)

Definition 2.2. [3] Let C and D be a SVNSs of V and E, respectively. Then D is said to be single valued neutrosophic relation (SVNR) on C, whenever TD (xy) ≤ min(TC (x), TC (y)) ID (xy) ≥ max(IC (x), IC (y)) FD (xy) ≥ max(FC (x), FC (y)) ∀x, y ∈ V. Definition 2.3. [4] Let C and D be IVNSs of a V and E, respectively. Then D is said to be interval valued neutrosophic relation (IVNR) on C, whenever TDL (xy) ≤ min(TCL (x), TCL (y)), IDL (xy) ≥ max(ICL (x), ICL (y)) FDL (xy) ≥ max(FCL (x), FCL (y)), TDU (xy) ≤ min(TCU (x), TCU (y)) IDU (xy) ≥ max(ICU (x), ICU (y)), FDU (xy) ≥ max(FCU (x), FCU (y)) ∀x, y ∈ V. Definition 2.4. [4] The interval valued neutrosophic graph (IVNG) is a pair G = (C, D) of G∗ = (V, E), where C is IVNS on V and D is IVNS on E, such that TDL (αβ) ≤ min(TCL (α), TCL (β)), IDL (αβ) ≥ max(ICL (α), ICL (β)) FDL (αβ) ≥ max(FCL (α), FCL (β)), TDU (αβ) ≤ min(TCU (α), TCU (β)) IDU (αβ) ≥ max(ICU (α), ICU (β)), FDU (αβ) ≥ max(FCU (α), FCU (β)) whenever 0 ≤ TDL (αβ) + IDL (αβ) + FDL (αβ) ≤ 3 0 ≤ TDU (αβ) + IDU (αβ) + FDU (αβ) ≤ 3 ∀ α, β ∈ V. The IVNG G is said to be complete (strong) IVNG, if TDL (xy) = min(TCL (x), TCL (y)), IDL (xy) = max(ICL (x), ICL (y)) FDL (xy) = max(FCL (x), FCL (y)), TDU (xy) = min(TCU (x), TCU (y)) IDU (xy) = max(ICU (x), ICU (y)), FDU (xy) = max(FCU (x), FCU (y)) ∀ x, y ∈ V (∀xy ∈ E). The order and size of G and also degree of vertex defined below O(G) = ([OT L (G), OT U (G)], [OIL (G), OIU (G)], [OF L (G), OF U (G)]) where OT L (G) =

X

OT U (G) =

X

TCL (α), OIL (G) =

α∈V

X α∈V

TCU (α), OIU (G) =

α∈V

X

X

ICL (α), OF L (G) =

FCL (α),

α∈V

X

ICU (α), OF U (G) =

α∈V

FCU (α).

α∈V

S(G) = ([ST L (G), ST U (G)], [SIL (G), SIU (G)], [SF L (G), SF U (G)]) where ST L (G) =

X

TDL (αβ), SIL (G) =

αβ∈E

ST U (G) =

X

αβ∈E

X

IDL (αβ), SF L (G) =

αβ∈E

TDU (αβ), SIU (G) =

X

αβ∈E

X

FDL (αβ),

αβ∈E

IDU (αβ), SF U (G) =

X

αβ∈E

FDU (αβ).

M. A. MALIK, A. HASSAN

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dG (α), is defined by dG (α) = ([dT L (α), dT U (α)], [dIL (α), dIU (α)], [dF L (α), dF U (α)]), where dT L (α) =

X

TDL (αβ), dIL (α) =

αβ∈E

dT U (α) =

X

αβ∈E

X

IDL (αβ), dF L (α) =

αβ∈E

TDU (αβ), dIU (α) =

X

X

FDL (αβ),

αβ∈E

IDU (αβ), dF U (α) =

αβ∈E

X

FDU (αβ).

αβ∈E

3. Special Types of IVNGs Definition 3.1. Let G1 = (C1 , D1 ) and G2 = (C2 , D2 ) be two IVNGs of G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ), respectively. Then the homomorphism χ : V 1 → V2 is a mapping from V1 into V2 satisfying following conditions TC1 L (p) ≤ TC2 L (χ(p)), IC1 L (p) ≥ IC2 L (χ(p)), FC1 L (p) ≥ FC2 L (χ(p)) TC1 U (p) ≤ TC2 U (χ(p)), IC1 U (p) ≥ IC2 U (χ(p)), FC1 U (p) ≥ FC2 U (χ(p)) ∀p ∈ V1 .

TD1 L (pq) ≤ TD2 L (χ(p)χ(q)), ID1 L (pq) ≥ ID2 L (χ(p)χ(q)), FD1 L (pq) ≥ FD2 L (χ(p)χ(q)) TD1 U (pq) ≤ TD2 U (χ(p)χ(q)), ID1 U (pq) ≥ ID2 U (χ(p)χ(q)), FD1 U (pq) ≥ FD2 U (χ(p)χ(q)) ∀pq ∈ E1 . The weak isomorphism υ : V1 → V2 is a bijective homomorphism from V1 into V2 satisfying following conditions TC1 L (p) = TC2 L (υ(p)), IC1 L (p) = IC2 L (υ(p)), FC1 L (p) = FC2 L (υ(p)) TC1 U (p) = TC2 U (υ(p)), IC1 U (p) = IC2 U (υ(p)), FC1 U (p) = FC2 U (υ(p)) ∀p ∈ V1 . The co-weak isomorphism κ : V1 → V2 is a bijective homomorphism from V1 into V2 satisfying following conditions TD1 L (pq) = TD2 L (κ(p)κ(q)), ID1 L (pq) = ID2 L (κ(p)κ(q)), FD1 L (pq) = FD2 L (κ(p)κ(q)) TD1 U (pq) = TD2 U (κ(p)κ(q)), ID1 U (pq) = ID2 U (κ(p)κ(q)), FD1 U (pq) = FD2 U (κ(p)κ(q)) ∀pq ∈ E1 . An isomorphism ψ : V1 → V2 is a bijective homomorphism from V1 into V2 satisfying following conditions TC1 L (p) = TC2 L (ψ(p)), IC1 L (p) = IC2 L (v(p)), FC1 L (p) = FC2 L (ψ(p)) TC1 U (p) = TC2 U (ψ(p)), IC1 U (p) = IC2 U (v(p)), FC1 U (p) = FC2 U (ψ(p)) ∀p ∈ V1 .

TD1 L (pq) = TD2 L (ψ(p)ψ(q)), ID1 L (pq) = ID2 L (ψ(p)ψ(q)), FD1 L (pq) = FD2 L (ψ(p)ψ(q)) TD1 U (pq) = TD2 U (ψ(p)ψ(q)), ID1 U (pq) = ID2 U (ψ(p)ψ(q)), FD1 U (pq) = FD2 U (ψ(p)ψ(q)) ∀pq ∈ E1 .

Remark 3.1. The weak isomorphism between two IVNGs preserves the orders.

Remark 3.2. The weak isomorphism between IVNGs is a partial order relation.

Remark 3.3. The co-weak isomorphism between two IVNGs preserves the sizes.

Remark 3.4. The co-weak isomorphism between IVNGs is a partial order relation.

Remark 3.5. The isomorphism between two IVNGs is an equivalence relation.

Remark 3.6. The isomorphism between two IVNGs preserves the orders and sizes.

4

Figure 1: Crisp Graph of IVNG.

Remark 3.7. The isomorphism between two IVNGs preserves the degrees of their vertices.

Definition 3.2. The subdivision IVNG sd(G) = (C, D) of IVNG G = (A, B), where C is a IVNS on V ∪ E and D is a IVNR on C, such that (1) C = A on V and C = B on E. (2) If v ∈ V lie on edge e ∈ E, then TDL (ve) = min(TAL (v), TBL (e)), IDL (ve) = max(IAL (v), IBL (e)) FDL (ve) = max(FAL (v), FBL (e)), TDU (ve) = min(TAU (v), TBU (e)) IDU (ve) = max(IAU (v), IBU (e)), FDU (ve) = max(FAU (v), FBU (e)) else D(ve) = O = ([0, 0], [0, 0], [0, 0]).

Proposition 3.1. Let G be a IVNG and sd(G) be the subdivision IVNG of a IVNG G, then (1) O(sd(G)) = O(G) + S(G).

(2) S(sd(G)) = 2S(G).

Proposition 3.2. If G is complete IVNG, then sd(G) need not to be complete IVNG. Example 3.1. Consider the crisp graph G ∗ = (V, E) of IVNG G = (A, B), which is shown in Figure 1. The IVNSs A and B over V = {a, b, c} and E = {p = ab, q = bc, r = ac}, which are defined in Table 1.

A TA IA FA B TB IB FB a [0.2,0.3] [0.1,0.2] [0.4,0.5] p [0.2,0.3] [0.4,0.5] [0.5,0.6] b [0.3,0.4] [0.2,0.3] [0.5,0.6] q [0.3,0.4] [0.8,0.9] [0.6,0.7] c [0.4,0.5] [0.7,0.8] [0.6,0.7] r [0.1,0.2] [0.7,0.8] [0.9,1.0] Table 1: IVNSs of IVNG. The crisp graph of SDIVNG sd(G) = (C, D) of a IVNG G, which is shown in Figure 2.

By calculations the IVNSs C and D, which are defined in Table 2.

M. A. MALIK, A. HASSAN

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q

c

r

b

p a Figure 2: Crisp Graph of SDIVNG.

C a p b q c r

TC [0.2,0.3] [0.2,0.3] [0.3,0.4] [0.3,0.4] [0.4,0.5] [0.1,0.2]

IC [0.1,0.2] [0.4,0.5] [0.2,0.3] [0.8,0.9] [0.7,0.8] [0.7,0.8]

FC D TD ID FD [0.4,0.5] ap [0.2,0.3] [0.4,0.5] [0.5,0.6] [0.5,0.6] pb [0.2,0.3] [0.4,0.5] [0.5,0.6] [0.5,0.6] bq [0.3,0.4] [0.8,0.9] [0.6,0.7] [0.6,0.7] qc [0.3,0.4] [0.8,0.9] [0.6,0.7] [0.6,0.7] cr [0.1,0.2] [0.7,0.8] [0.9,1.0] [0.9,1.0] ra [0.1,0.2] [0.7,0.8] [0.9,1.0]

Table 2: IVNSs of SDIVNG.

Definition 3.3. The total interval valued neutrosophic graph (TIVNG) T (G) = (C, D) of G = (A, B), where C is a IVNS on V ∪ E and D is a IVNR on C, such that (1) C = A on V and C = B on E. (2) If v ∈ V lie on edge e ∈ E, then TDL (ve) = min(TAL (v), TBL (e)), IDL (ve) = max(IAL (v), IBL (e)) FDL (ve) = max(FAL (v), FBL (e)), TDU (ve) = min(TAU (v), TBU (e)) IDU (ve) = max(IAU (v), IBU (e)), FDU (ve) = max(FAU (v), FBU (e)) else D(ve) = O = ([0, 0], [0, 0], [0, 0]).

(3) If α, β ∈ E, then TDL (αβ) = TBL (αβ), IDL (αβ) = IBL (αβ), FDL (αβ) = FBL (αβ), TDU (αβ) = TBU (αβ), IDU (αβ) = IBU (αβ), FDU (αβ) = FBU (αβ).

(4) If e, f ∈ E have a common vertex, then TDL (ef ) = min(TBL (e), TBL (f )), IDL (ef ) = max(IBL (e), IBL (f )) FDL (ef ) = max(FBL (e), FBL (f )), TDU (ef ) = min(TBU (e), TBU (f )) IDU (ef ) = max(IBU (e), IBU (f )), FDU (ef ) = max(FBU (e), FBU (f )) else D(ef ) = O = ([0, 0], [0, 0], [0, 0]).

6

r

c

p

a

q

b Figure 3: Crisp Graph of TIVNG.

D ab bc ca pq qr rp

TD [0.2,0.3] [0.3,0.4] [0.1,0.2] [0.2,0.3] [0.1,0.2] [0.1,0.2]

ID [0.4,0.5] [0.8,0.9] [0.7,0.8] [0.8,0.9] [0.8,0.9] [0.7,0.8]

FD D TD ID FD [0.5,0.6] ap [0.2,0.3] [0.4,0.5] [0.5,0.6] [0.6,0.7] pb [0.2,0.3] [0.4,0.5] [0.5,0.6] [0.9,1.0] bq [0.3,0.4] [0.8,0.9] [0.6,0.7] [0.6,0.7] qc [0.3,0.4] [0.8,0.9] [0.6,0.7] [0.9,1.0] cr [0.1,0.2] [0.7,0.8] [0.9,1.0] [0.9,1.0] ra [0.1,0.2] [0.7,0.8] [0.9,1.0]

Table 3: IVNS of TIVNG. Example 3.2. In Example 3.1, the crisp graph for TIVNG T (G) = (C, D), which is shown in Figure 3. Here C is defined in Example 3.1. By calculations the IVNS D, which is defined in Table 3.

Proposition 3.3. Let G be a IV N G and T (G) be the TIVNG of G, then (1) O(T (G)) = O(G) + S(G) = O(sd(G)).

(2) S(sd(G)) = 2S(G).

Proposition 3.4. If G is a IVNG, then sd(G) is weak isomorphic to T (G).

Definition 3.4. The middle interval valued neutrosophic graph (MIVNG) M (G) = (C, D) of G = (A, B), where C is a IVNS on V ∪ E and D is a IVNR on C, such that (1) C = A on V and C = B on E, else C = O = ([0, 0], [0, 0], [0, 0]).

(2) If v ∈ V lie on edge e ∈ E, then TDL (ve) = TBL (e), IDL (ve) = IBL (e), FDL (ve) = FBL (e) TDU (ve) = TBU (e), IDU (ve) = IBU (e), FDU (ve) = FBU (e) else D(ve) = O = ([0, 0], [0, 0], [0, 0]).

(3) If u, v ∈ V, then D(uv) = O = ([0, 0], [0, 0], [0, 0]).

(4) If e, f ∈ E such that e and f are adjacent in G, then TDL (ef ) = TBL (uv), IDL (ef ) = IBL (uv), FDL (ef ) = FBL (uv), TDU (ef ) = TBU (uv), IDU (ef ) = IBU (uv), FDU (ef ) = FBU (uv).

Remark 3.8. If G is a IVNG and M (G) is a MIVNG of G, then O(M (G)) = O(G) + S(G).

M. A. MALIK, A. HASSAN

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b

e1

e2

c

a

Figure 4: Crisp Graph of IVNG.

a

b

c

e2

e1

Figure 5: Crisp Graph of MIVNG.

Remark 3.9. If G is a IVNG, then M (G) is a strong IVNG. Remark 3.10. If G is complete IVNG, then M (G) need not to be complete IVNG. Example 3.3. Consider the IVNG G = (A, B) of a G ∗ , which is shown in Figure 4. The IVNSs A and B are defined in Table 4. The crisp graph of MIVNG M (G) = (C, D), which is shown in Figure 5. By calculations, the IVNSs C and D are defined in Table 5.

A a b c B e1 e2

TA [0.3,0.4] [0.7,0.8] [0.9,1.0] TB [0.2,0.3] [0.4,0.5]

IA [0.4,0.5] [0.6,0.7] [0.7,0.8] IB [0.6,0.7] [0.8,0.9]

FA [0.4,0.5] [0.3,0.4] [0.2,0.3] FB [0.6,0.7] [0.7,0.8]

Table 4: IVNSs of IVNG.

C a b c e1 e2

TC [0.3,0.4] [0.7,0.8] [0.9,1.0] [0.2,0.3] [0.4,0.5]

IC [0.4,0.5] [0.6,0.7] [0.7,0.8] [0.6,0.7] [0.8,0.9]

FC D [0.5,0.6] e1 e2 [0.3,0.4] ae 1 [0.2,0.3] be 1 [0.6,0.7] be2 [0.7,0.8] ce2

TD [0.2,0.3] [0.2,0.3] [0.2,0.3] [0.2,0.3] [0.4,0.5]

ID [0.8,0.9] [0.6,0.7] [0.6,0.7] [0.6,0.7] [0.8,0.9]

FD [0.7,0.8] [0.6,0.7] [0.6,0.7] [0.6,0.7] [0.7,0.8]

8

Table 5: IVNSs of MIVNG. Proposition 3.5. If G is a IVNG, then sd(G) is weak isomorphic with M (G).

Proposition 3.6. If G is a IVNG, then M (G) is weak isomorphic with T (G).

Proposition 3.7. If G is a IVNG, then T (G) is isomorphic with G ∪ M (G).

Definition 3.5. Let the intersection graph be P (X) = (X, Y ) of a G ∗ , let C1 and D1 be IVNSs over V and E. Also let C2 and D2 be IVNSs over X and Y. Then the interval valued neutrosophic intersection graph (IVNIG) of a IVNG G = (C 1 , D1 ) is a IVNG P (G) = (C2 , D2 ), such that TC2 L (Xi ) = TC1 L (vi ), IC2 L (Xi ) = IC1 L (vi ), FC2 L (Xi ) = FC1 L (vi ) TC2 U (Xi ) = TC1 U (vi ), IC2 U (Xi ) = IC1 U (vi ), FC2 U (Xi ) = FC1 U (vi ) TD2 L (Xi Xj ) = TD1 L (vi vj ), ID2 L (Xi Xj ) = ID1 L (vi vj ), FD2 L (Xi Xj ) = FD1 L (vi vj ) TD2 U (Xi Xj ) = TD1 U (vi vj ), ID2 U (Xi Xj ) = ID1 U (vi vj ), FD2 U (Xi Xj ) = FD1 U (vi vj ) ∀ Xi , Xj ∈ X and Xi Xj ∈ Y. Proposition 3.8. Let G = (A1 , B1 ) be a IVNG of G∗ = (V, E) and let P (G) = (A2 , B2 ) be a IVNIG, then (1) The IVNIG is a IVNG. (2) The IVNG is isomorphic to IVNIG. Proof. (1) By the definition of IVNIG TB2 L (Si Sj ) IB2 L (Si Sj ) FB2 L (Si Sj ) TB2 U (Si Sj ) IB2 U (Si Sj ) FB2 U (Si Sj )

= = = = = =

TB1 L (vi vj ) ≤ min(TA1 L (vi ), TA1 L (vj )) = min(TA2 L (Si ), TA2 L (Sj )) IB1 L (vi vj ) ≥ max(IA1 L (vi ), IA1 L (vj )) = max(IA2 L (Si ), IA2 L (Sj )) FB1 L (vi vj ) ≥ max(FA1 L (vi ), FA1 L (vj )) = max(FA2 L (Si ), FA2 L (Sj )) TB1 U (vi vj ) ≤ min(TA1 U (vi ), TA1 U (vj )) = min(TA2 U (Si ), TA2 U (Sj )) IB1 U (vi vj ) ≥ max(IA1 U (vi ), IA1 U (vj )) = max(IA2 U (Si ), IA2 U (Sj )) FB1 U (vi vj ) ≥ max(FA1 U (vi ), FA1 U (vj )) = max(FA2 U (Si ), FA2 U (Sj ))

this shows that IVNIG is a IVNG. (2) Define f : V → X by f (vi ) = Si for i = 1, 2, 3, . . . , n clearly f is bijective. Next vi vj ∈ E if and only if Si Sj ∈ T and T = {f (vi )f (vj ) : vi vj ∈ E}, also TA2 L (f (vi )) = TA2 L (Si ) = TA1 L (vi ), IA2 L (f (vi )) = IA2 L (Si ) = IA1 L (vi ) FA2 L (f (vi )) = FA2 L (Si ) = FA1 L (vi ), TA2 U (f (vi )) = TA2 U (Si ) = TA1 U (vi ) IA2 U (f (vi )) = IA2 U (Si ) = IA1 U (vi ), FA2 U (f (vi )) = FA2 U (Si ) = FA1 U (vi ) ∀ vi ∈ V. TB2 L (f (vi )f (vj )) = TB2 L (Si Sj ) = TB1 L (vi vj ) IB2 L (f (vi )f (vj )) = IB2 L (Si Sj ) = IB1 L (vi vj ) FB2 L (f (vi )f (vj )) = FB2 L (Si Sj ) = FB1 L (vi vj ) TB2 U (f (vi )f (vj )) = TB2 U (Si Sj ) = TB1 U (vi vj ) IB2 U (f (vi )f (vj )) = IB2 U (Si Sj ) = IB1 U (vi vj ) FB2 U (f (vi )f (vj )) = FB2 U (Si Sj ) = FB1 U (vi vj ) ∀ vi vj ∈ E.

M. A. MALIK, A. HASSAN

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Definition 3.6. Let G∗ = (V, E) and L(G∗ ) = (X, Y ) be its line graph, where A1 and B1 be IVNSs over V and E. Let A2 and B2 be IVNSs over X and Y. The interval valued neutrosophic line graph (IVNLG) of IVNG G = (A 1 , B1 ) is IVNG L(G) = (A2 , B2 ), such that TA2 L (Sx ) = TB1 L (x) = TB1 L (ux vx ), IA2 L (Sx ) = IB1 L (x) = IB1 L (ux vx ) FA2 L (Sx ) = FB1 L (x) = FB1 L (ux vx ), TA2 U (Sx ) = TB1 U (x) = TB1 U (ux vx ) IA2 U (Sx ) = IB1 U (x) = IB1 U (ux vx ), FA2 U (Sx ) = FB1 U (x) = FB1 U (ux vx ) ∀ Sx , Sy ∈ X and TB2 L (Sx Sy ) = min(TB1 L (x), TB1 L (y)), IB2 L (Sx Sy ) = max(IB1 L (x), IB1 L (y)) FB2 L (Sx Sy ) = max(FB1 L (x), FB1 L (y)), TB2 U (Sx Sy ) = min(TB1 U (x), TB1 U (y)) IB2 U (Sx Sy ) = max(IB1 U (x), IB1 U (y)), FB2 U (Sx Sy ) = max(FB1 U (x), FB1 U (y)) ∀ Sx Sy ∈ Y. Example 3.4. Consider the G∗ = (V, E), where V = {α1 , α2 , α3 , α4 } and E = {x1 = α1 α2 , x2 = α2 α3 , x3 = α3 α4 , x4 = α4 α1 } and G = (A1 , B1 ) be a IVNG of G∗ = (V, E), which is defined in in Table 6. Consider the L(G ∗ ) = (X, Y ), such that X = {Γx1 , Γx2 , Γx3 , Γx4 } and Y = {Γx1 Γx2 , Γx2 Γx3 , Γx3 Γx4 , Γx4 Γx1 }. Let A2 and B2 be IVNSs over X and Y. Then by calculations, IVNLG L(G) is defined in Table 7.

A1 α1 α2 α3 α4

T A1 [0.2,0.3] [0.4,0.5] [0.4,0.5] [0.3,0.4]

I A1 [0.5,0.6] [0.3,0.4] [0.5,0.6] [0.2,0.3]

F A1 B1 [0.5,0.6] x1 [0.3,0.4] x2 [0.5,0.6] x3 [0.2,0.3] x4

T B1 [0.1,0.2] [0.3,0.4] [0.2,0.3] [0.1,0.2]

I B1 [0.6,0.7] [0.6,0.7] [0.7,0.8] [0.7,0.8]

F B1 [0.7,0.8] [0.7,0.8] [0.8,0.9] [0.8,0.9]

Table 6: IVNSs of IVNG. A2 Γ x1 Γ x2 Γ x3 Γ x4

T A2 [0.1,0.2] [0.3,0.4] [0.2,0.3] [0.1,0.2]

I A2 [0.6,0.7] [0.6,0.7] [0.7,0.8] [0.7,0.8]

F A2 [0.7,0.8] [0.7,0.8] [0.8,0.9] [0.8,0.9]

B2 Γ x1 Γ x2 Γ x2 Γ x3 Γ x3 Γ x4 Γ x4 Γ x1

T B2 [0.1,0.2] [0.2,0.3] [0.1,0.2] [0.1,0.2]

I B2 [0.6,0.7] [0.7,0.8] [0.7,0.8] [0.7,0.8]

F B2 [0.7,0.8] [0.8,0.9] [0.8,0.9] [0.8,0.9]

Table 7: IVNSs of IVNLG. Proposition 3.9. Every IVNLG is a strong IVNG. Proposition 3.10. The L(G) = (A2 , B2 ) is a IVNLG corresponding to IVNG G = (A1 , B1 ).

Proposition 3.11. The L(G) = (A2 , B2 ) is a IVNLG of some IVNG G = (A1 , B1 ) if and only if TB2 L (Sx Sy ) = min(TA2 L (Sx ), TA2 L (Sy )), IB2 L (Sx Sy ) = max(IA2 L (Sx ), IA2 L (Sy )) FB2 L (Sx Sy ) = max(FA2 L (Sx ), FA2 L (Sy )), TB2 U (Sx Sy ) = min(TA2 U (Sx ), TA2 U (Sy )) IB2 U (Sx Sy ) = max(IA2 U (Sx ), IA2 U (Sy )), FB2 U (Sx Sy ) = max(FA2 U (Sx ), FA2 U (Sy )) ∀ Sx Sy ∈ Y.

10

Proof. Assume that TB2 L (Sx Sy ) = min(TA2 L (Sx ), TA2 L (Sy )), IB2 L (Sx Sy ) = max(IA2 L (Sx ), IA2 L (Sy )) FB2 L (Sx Sy ) = max(FA2 L (Sx ), FA2 L (Sy )), TB2 U (Sx Sy ) = min(TA2 U (Sx ), TA2 U (Sy )) IB2 U (Sx Sy ) = max(IA2 U (Sx ), IA2 U (Sy )), FB2 U (Sx Sy ) = max(FA2 U (Sx ), FA2 U (Sy )) ∀ Sx Sy ∈ Y. Next define TA1 L (x) = TA2 L (Sx ), IA1 L (x) = IA2 L (Sx ), FA1 L (x) = FA2 L (Sx ) TA1 U (x) = TA2 U (Sx ), IA1 U (x) = IA2 U (Sx ), FA1 U (x) = FA2 U (Sx ) ∀ x ∈ E, then TB2 L (Sx Sy ) = min(TA2 L (Sx ), TA2 L (Sy )) = min(TA2 L (x), TA2 L (y)) IB2 L (Sx Sy ) = max(IA2 L (Sx ), IA2 L (Sy )) = max(IA2 L (x), IA2 L (y)) FB2 L (Sx Sy ) = max(FA2 L (Sx ), FA2 L (Sy )) = max(FA2 L (x), FA2 L (y)) TB2 U (Sx Sy ) = min(TA2 U (Sx ), TA2 U (Sy )) = min(TA2 U (x), TA2 U (y)) IB2 U (Sx Sy ) = max(IA2 U (Sx ), IA2 U (Sy )) = max(IA2 U (x), IA2 U (y)) FB2 U (Sx Sy ) = max(FA2 U (Sx ), FA2 U (Sy )) = max(FA2 U (x), FA2 U (y)) The IVNS A1 that yields the property TB1 L (xy) ≤ min(TA1 L (x), TA1 L (y)), IB1 L (xy) ≥ max(IA1 L (x), IA1 L (y)) FB1 L (xy) ≥ max(FA1 L (x), FA1 L (y)), TB1 U (xy) ≤ min(TA1 U (x), TA1 U (y)) IB1 U (xy) ≥ max(IA1 U (x), IA1 U (y)), FB1 U (xy) ≥ max(FA1 U (x), FA1 U (y)) will suffice. Converse is straight forward.

Proposition 3.12. If L(G) = (A2 , B2 ) is IVNLG of IVNG G = (A1 , B1 ), then L(G∗ ) is the crisp line graph of G∗ .

Proof. Since L(G) be a IVNLG, TA2 L (Sx ) = TB1 L (x), IA2 L (Sx ) = IB1 L (x), FA2 L (Sx ) = FB1 L (x) ∀ x ∈ E and so Sx ∈ X if and only if x ∈ E, also TB2 L (Sx Sy ) = min(TB1 L (x), TB1 L (y)), IB2 L (Sx Sy ) = max(IB1 L (x), IB1 L (y)) FB2 L (Sx Sy ) = max(FB1 L (x), FB1 L (y)), TB2 U (Sx Sy ) = min(TB1 U (x), TB1 U (y)) IB2 U (Sx Sy ) = max(IB1 U (x), IB1 U (y)), FB2 U (Sx Sy ) = max(FB1 U (x), FB1 U (y)) ∀ Sx Sy ∈ Y and so, Y = {Sx Sy : Sx ∩ Sy 6= φ, x, y ∈ E, x 6= y}.

Proposition 3.13. If L(G) = (A2 , B2 ) is IVNLG of IVNG G = (A1 , B1 ) if and only if L(G∗ ) = (X, Y ) is the line graph and TB2 L (xy) = min(TA2 L (x), TA2 L (y)), IB2 L (xy) = max(IA2 L (x), IA2 L (y)) FB2 L (xy) = max(FA2 L (x), FA2 L (y)), TB2 U (xy) = min(TA2 U (x), TA2 U (y)) IB2 U (xy) = max(IA2 U (x), IA2 U (y)), FB2 U (xy) = max(FA2 U (x), FA2 U (y)) ∀ xy ∈ Y. Proof. It follows from Propositions 3.11 and 3.12.

Proposition 3.14. Let G be a IVNG, then M (G) is isomorphic with sd(G) ∪ L(G).

M. A. MALIK, A. HASSAN

11

Theorem 3.1. Let L(G) = (A2 , B2 ) be IVNLG corresponding to IVNG G = (A 1 , B1 ). (a) If G is weak isomorphic onto L(G) if and only if ∀ v ∈ V, x ∈ E and G ∗ to be a cycle, such that TA1 L (v) = TB1 L (x), IA1 L (v) = IB1 L (x), FA1 L (v) = TB1 L (x), TA1 U (v) = TB1 U (x), IA1 U (v) = IB1 U (x), FA1 U (v) = TB1 U (x). (b) If G is weak isomorphic onto L(G), then G and L(G) are isomorphic.

Proof. By hypothesis G∗ is a cycle. Let V = {v1 , v2 , v3 , . . . , vn } and E = {x1 = v1 v2 , x2 = v2 v3 , . . . , xn = vn v1 } where P : v1 v2 v3 . . . vn is a cycle, characterize a IVNS A1 by A1 (vi ) = 0 0 0 0 0 0 ([pi , pi ], [qi , qi ], [ri , ri ]) and B1 by B1 (xi ) = ([ai , ai ], [bi , bi ], [ci , ci ]) for i = 1, 2, 3, . . . , n and 0 0 0 0 0 0 vn+1 = v1 , if pn+1 = p1 , qn+1 = q1 , rn+1 = r1 , pn+1 = p1 , qn+1 = q1 , rn+1 = r1 . Thus ai ≤ min(pi , pi+1 ), bi ≥ max(qi , qi+1 ), ci ≥ max(ri , ri+1 ) 0

0

0

0

0

0

0

0

0

0

0

ai ≤ min(pi , pi+1 ), bi ≥ max(qi , qi+1 ), ci ≥ max(ri , ri+1 ) for i = 1, 2, 3, . . . , n. Next X = {Γx1 , Γx2 , . . . , Γxn } and Y = {Γx1 Γx2 , Γx2 Γx3 , . . . , Γxn Γx1 }, 0 0 0 0 0 0 thus for an+1 = a1 , an+1 = a1 , bn+1 = b1 , bn+1 = b1 , cn+1 = c1 , cn+1 = c1 to obtain 0

A2 (Γxi ) = B1 (xi ) = ([ai , ai ], [bi , bi ], [ci , ci ]) 0

0

0

0

and B2 (Γxi Γxi+1 ) = ([min(ai , ai+1 ), min(ai , ai+1 )], [max(bi , bi+1 ), max(bi , bi+1 )], [max(ci , ci+1 ), 0 0 max(ci , ci+1 )]) for i = 1, 2, 3, . . . , n and vn+1 = v1 . Since f preserves adjacency, hence it induce permutation π of {1, 2, 3, . . . , n}, f (v i ) = Γvπ(i) vπ(i)+1 and vi vi+1 → f (vi )f (vi+1 ) = Γvπ(i) vπ(i)+1 Γvπ(i+1) vπ(i+1)+1 for i = 1, 2, 3, . . . , n − 1. Therefore pi = TA1 L (vi ) ≤ TA2 L (f (vi )) = TA2 L (Γvπ(i) vπ(i)+1 ) = TB1 L (vπ(i) vπ(i)+1 ) = aπ(i) 0

0

0

0

Similarly, pi ≤ aπ(i) , qi ≥ bπ(i) , ri ≥ cπ(i) , qi ≥ bπ(i) , ri ≥ cπ(i) and ai = TB1 L (vi vi+1 ) ≤ TB2 L (f (vi )f (vi+1 )) = TB2 L (Γvπ(i) vπ(i)+1 Γvπ(i+1) vπ(i+1)+1 ) = min(TB1 L (vπ(i) vπ(i)+1 ), TB1 L (vπ(i+1) vπ(i+1)+1 )) = min(aπ(i) , aπ(i)+1 ) 0

0

0

0

0

0

similarly ai ≤ min(aπ(i) , aπ(i)+1 ), bi ≥ max(bπ(i) , bπ(i)+1 ), bi ≥ max(bπ(i) , bπ(i)+1 ) and 0

0

0

ci ≥ max(cπ(i) , cπ(i)+1 ), ci ≥ max(cπ(i) , cπ(i)+1 ) for i = 1, 2, 3, . . . , n. Therefore 0

0

0

0

0

0

pi ≤ aπ(i) , qi ≥ bπ(i) , ri ≥ cπ(i) , pi ≤ aπ(i) , qi ≥ bπ(i) , ri ≥ cπ(i) and ai ≤ min(aπ(i) , aπ(i)+1 ), bi ≥ max(bπ(i) , bπ(i)+1 ), ci ≥ max(cπ(i) , cπ(i)+1 ) 0

0

0

0

0

0

0

0

0

ai ≤ min(aπ(i) , aπ(i)+1 ), bi ≥ max(bπ(i) , bπ(i)+1 ), ci ≥ max(cπ(i) , cπ(i)+1 ) thus

0

0

0

0

0

ai ≤ aπ(i) , bi ≥ bπ(i) , ci ≥ cπ(i) , ai ≤ aπ(i) , bi ≥ bπ(i) , ci ≥ cπ(i) and so aπ(i) ≤ aπ(π(i)) , bπ(i) ≥ bπ(π(i)) , cπ(i) ≥ cπ(π(i)) 0

0

0

0

0

0

aπ(i) ≤ aπ(π(i)) , bπ(i) ≥ bπ(π(i)) , cπ(i) ≥ cπ(π(i)) ∀ i = 1, 2, 3, . . . , n. Next to extend ai ≤ aπ(i) ≤ . . . ≤ aπj (i) ≤ ai , bi ≥ bπ(i) ≥ . . . ≥ bπj (i) ≥ bi

12 0

0

0

0

0

0

0

ci ≥ cπ(i) ≥ . . . ≥ cπj (i) ≥ ci , ai ≤ aπ(i) ≤ . . . ≤ aπj (i) ≤ ai 0

0

0

0

0

bi ≥ bπ(i) ≥ . . . ≥ bπj (i) ≥ bi , ci ≥ cπ(i) ≥ . . . ≥ cπj (i) ≥ ci where π j+1 identity. Hence 0

0

0

0

0

0

ai = aπ(i) , bi = bπ(i) , ci = cπ(i) , ai = aπ(i) , bi = bπ(i) , ci = cπ(i) ∀ i = 1, 2, 3, . . . , n. Therefore ai ≤ aπ(i+1) = ai+1 , bi ≥ bπ(i+1) = bi+1 , ci ≥ cπ(i+1) = ci+1 0

0

0

0

0

0

0

0

0

ai ≤ aπ(i+1) = ai+1 , bi ≥ bπ(i+1) = bi+1 , ci ≥ cπ(i+1) = ci+1 which together with 0

0

0

0

0

0

an+1 = a1 , bn+1 = b1 , cn+1 = c1 , an+1 = a1 , bn+1 = b1 , cn+1 = c1 which implies that 0

0

0

0

0

0

ai = a 1 , b i = b 1 , c i = c 1 , a i = a 1 , b i = b 1 , c i = c 1 ∀ i = 1, 2, 3, . . . , n. Thus a1 = a 2 = . . . = a n = p 1 = p 2 = . . . = p n 0

0

0

0

0

0

a1 = a 2 = . . . = a n = p 1 = p 2 = . . . = p n b1 = b 2 = . . . = b n = q 1 = q 2 = . . . = q n 0

0

0

0

0

0

0

0

0

0

0

0

b1 = b 2 = . . . = b n = q 1 = q 2 = . . . = q n c1 = c 2 = . . . = c n = r 1 = r 2 = . . . = r n c1 = c 2 = . . . = c n = r 1 = r 2 = . . . = r n Therefore (a) and (b) holds, since converse of result (a) is straight forward.

4. Conclusion The neutrosophic graphs have many applications in path problems, networks and computer science. Strong IVNG and complete IVNG are the types of IVNG. In this paper, we discussed the special types of IVNGs, subdivision IVNGs, middle IVNGs, total IVNGs and IVNLGs of the given IVNGs. We investigated isomorphism properties of subdivision IVNGs, middle IVNGs, total IVNGs and IVNLGs. References [1] H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued Neutrosophic Sets, Multisspace and Multistructure, (2010) 410-413.

[2] H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Interval neutrosophic sets and logic, Theory and Applications in Computing, Phoenix, AZ, USA Hexis, (2005).

[3] S. Broumi, M. Talea, A. Bakali and F. Smarandache, Single valued neutrosophic graphs, Journal of New Theory, (2016), [4] S. Broumi, M. Talea, A. Bakali and F. Smarandache, Interval valued neutrosophic graphs, Creighton University, Vol XII, 2016, pp 5-33.

[5] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters, Vol. 6 (1987).

[6] A. Rosenfeld, Fuzzy graphs, In Fuzzy Sets and Applications (Zadeh, L. A., K. S. Fu, M. Shimura, Eds.), Academic Press, New York, (1975).

[7] A. N. Gani and J. Malarvizhi, On antipodal fuzzy graph, Applied Mathematical Sciences, No. 4 (2010) 2145-2155.

[8] S. Broumi, M. Talea, A. Bakali and F. Smarandache, Applying Dijkstra algorithm for solving neutrosophic shortest path problem, Proceedings of the 2016 International Conference on Advanced Mechatronic Systems, Melbourne, Australia, (2016) 412-416.

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[9] S. Broumi, M. Talea, A. Bakali, F. Smarandache and M. Ali, Shortest path problem under bipolar neutrosphic setting, Applied Mechanics and Materials, vol. 859 (2016) 59-66.

[10] S. Broumi, M. Talea, A. Bakali, F. Smarandache and L. Vladareanu, Computation of shortest path problem in a network with SV-Trapezoidal neutrosophic numbers, Proceedings of the 2016 International Conference on Advanced Mechatronic Systems, Melbourne, Australia, (2016) 417-422.